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Beliebt Rechnen >

integral of (x^2-4)/(x(x^2-1)(x^2+1))

Lösung

\int \frac{x ^{2} - 4}{x ( x ^{2} - 1 ) ( x ^{2} + 1 )} d x

Lösung

4 ln ∣ x ∣ - \frac{3}{4} ln ∣ x + 1 ∣ - \frac{3}{4} ln ∣ x - 1 ∣ - \frac{5}{4} ln x^{2} + 1 + C

Schritte zur Lösung

\int \frac{x ^{2} - 4}{x ( x ^{2} - 1 ) ( x ^{2} + 1 )} d x

Ermittle den Partialbruch von

\frac{x ^{2} - 4}{x ( x ^{2} - 1 ) ( x ^{2} + 1 )} : \frac{4}{x} - \frac{3}{4 ( x + 1 )} - \frac{3}{4 ( x - 1 )} - \frac{5 x}{2 ( x ^{2} + 1 )}

= \int \frac{4}{x} - \frac{3}{4 ( x + 1 )} - \frac{3}{4 ( x - 1 )} - \frac{5 x}{2 ( x ^{2} + 1 )} d x

Wende die Summenregel an:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int \frac{4}{x} d x - \int \frac{3}{4 ( x + 1 )} d x - \int \frac{3}{4 ( x - 1 )} d x - \int \frac{5 x}{2 ( x ^{2} + 1 )} d x

\int \frac{4}{x} d x = 4 ln ∣ x ∣

\int \frac{3}{4 ( x + 1 )} d x = \frac{3}{4} ln ∣ x + 1 ∣

\int \frac{3}{4 ( x - 1 )} d x = \frac{3}{4} ln ∣ x - 1 ∣

\int \frac{5 x}{2 ( x ^{2} + 1 )} d x = \frac{5}{4} ln x^{2} + 1

= 4 ln ∣ x ∣ - \frac{3}{4} ln ∣ x + 1 ∣ - \frac{3}{4} ln ∣ x - 1 ∣ - \frac{5}{4} ln x^{2} + 1

Füge eine Konstante zur Lösung hinzu

= 4 ln ∣ x ∣ - \frac{3}{4} ln ∣ x + 1 ∣ - \frac{3}{4} ln ∣ x - 1 ∣ - \frac{5}{4} ln x^{2} + 1 + C

Graph

Beliebte Beispiele

integral of (3x+3)/(x^2+1)

\int \frac{3 x + 3}{x ^{2} + 1} d x

integral of 1/(u^2-5)

\int \frac{1}{u ^{2} - 5} d u

integral of (3+x^2)/x

\int \frac{3 + x ^{2}}{x} d x

integral of (x^2-6x+9)/(sqrt(x^2-6x+34))

\int \frac{x ^{2} - 6 x + 9}{x ^{2} - 6 x + 34} d x

integral of xsqrt(4-7x)

\int x 4 - 7 x d x